Finding the interval of convergence

Thread starter NihalRi
Start date Oct 27, 2016
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Convergence Interval

In summary, the interval of convergence is the range of values for the variable in a power series where the series will converge. It is determined by using the ratio test or the root test, with a limit of less than 1 indicating convergence within the interval. The interval can be infinite in either direction, and if the value of the variable is outside of the interval, the series will either diverge or have an indeterminate value. The interval of convergence can also change for different power series, depending on their coefficients and exponents.

Oct 27, 2016

NihalRi

Homework Statement

The question was to find the interval of convergence for a series.

Homework Equations

an+1/an

The Attempt at a Solution

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Oct 28, 2016

Mark44

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NihalRi said:

Homework Statement

The question was to find the interval of convergence for a series.

Homework Equations

an+1/an

The Attempt at a Solution

View attachment 108091

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Related to Finding the interval of convergence

1. What is the interval of convergence?

The interval of convergence is the range of values for the variable in a power series where the series will converge, or approach a finite value. It is typically represented as an interval, such as [-R, R], where R is the radius of convergence.

2. How is the interval of convergence determined?

The interval of convergence is determined by using the ratio test or the root test on the power series. The ratio test compares the absolute value of the ratio of consecutive terms in the series to a fixed value, while the root test compares the nth root of the absolute value of the terms to a fixed value. If the limit of either test is less than 1, the series will converge within the interval of convergence.

3. Can the interval of convergence be infinite?

Yes, the interval of convergence can be infinite in either direction. This means that the series will converge for all values of the variable, such as in the series for e^x, which has an interval of convergence of (-∞, ∞).

4. What happens if the value of the variable is outside of the interval of convergence?

If the value of the variable is outside of the interval of convergence, the series will either diverge or have an indeterminate value. This means that the series will not approach a finite value and may either increase or decrease without bound, or have a value that cannot be determined.

5. Can the interval of convergence change for different power series?

Yes, the interval of convergence can change for different power series. The radius of convergence, which determines the size of the interval, depends on the coefficients and exponents in the series. Therefore, different power series may have different intervals of convergence depending on their terms.